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Register a new userQuantum and Classical Strong Direct Product Theorems and Optimal Time-Space Tradeoffs
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Hartmut Klauck
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A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum query complexity of the OR function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing k instances of the Disjointness function. Our direct product theorems imply a time-space tradeoff T^2*S=Omega(N^3) for sorting N items on a quantum computer, which is optimal up to polylog factors. They also give several tight time-space and communication-space tradeoffs for the problems of Boolean matrix-vector multiplication and matrix multiplication.
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We give a new version of the adversary method for proving lower bounds on quantum query algorithms. The new method is based on analyzing the eigenspace structure of the problem at hand. We use it to prove a new and optimal strong direct product theorem for 2-sided error quantum algorithms computing k independent instances of a symmetric Boolean function: if the algorithm uses significantly less than k times the number of queries needed for one instance of the function, then its success probability is exponentially small in k. We also use the polynomial method to prove a direct product theorem for 1-sided error algorithms for k threshold functions with a stronger bound on the success probability. Finally, we present a quantum algorithm for evaluating solutions to systems of linear inequalities, and use our direct product theorems to show that the time-space tradeoff of this algorithm is close to optimal.
In function inversion, we are given a function $f: [N] mapsto [N]$, and want to prepare some advice of size $S$, such that we can efficiently invert any image in time $T$. This is a well studied problem with profound connections to cryptography, data structures, communication complexity, and circuit lower bounds. Investigation of this problem in the quantum setting was initiated by Nayebi, Aaronson, Belovs, and Trevisan (2015), who proved a lower bound of $ST^2 = tildeOmega(N)$ for random permutations against classical advice, leaving open an intriguing possibility that Grovers search can be sped up to time $tilde O(sqrt{N/S})$. Recent works by Hhan, Xagawa, and Yamakawa (2019), and Chung, Liao, and Qian (2019) extended the argument for random functions and quantum advice, but the lower bound remains $ST^2 = tildeOmega(N)$. In this work, we prove that even with quantum advice, $ST + T^2 = tildeOmega(N)$ is required for an algorithm to invert random functions. This demonstrates that Grovers search is optimal for $S = tilde O(sqrt{N})$, ruling out any substantial speed-up for Grovers search even with quantum advice. Further improvements to our bounds would imply new classical circuit lower bounds, as shown by Corrigan-Gibbs and Kogan (2019). To prove this result, we develop a general framework for establishing quantum time-space lower bounds. We further demonstrate the power of our framework by proving quantum time-space lower bounds for Yaos box problem and salted cryptography.
We construct a quantum oracle relative to which $mathsf{BQP} = mathsf{QMA}$ but cryptographic pseudorandom quantum states and pseudorandom unitary transformations exist, a counterintuitive result in light of the fact that pseudorandom states can be broken by quantum Merlin-Arthur adversaries. We explain how this nuance arises as the result of a distinction between algorithms that operate on quantum and classical inputs. On the other hand, we show that some computational complexity assumption is needed to construct pseudorandom states, by proving that pseudorandom states do not exist if $mathsf{BQP} = mathsf{PP}$. We discuss implications of these results for cryptography, complexity theory, and quantum tomography.
We give a direct product theorem for the entanglement-assisted interactive quantum communication complexity of an $l$-player predicate $mathsf{V}$. In particular we show that for a distribution $p$ that is product across the input sets of the $l$ players, the success probability of any entanglement-assisted quantum communication protocol for computing $n$ copies of $mathsf{V}$, whose communication is $o(log(mathrm{eff}^*(mathsf{V},p))cdot n)$, goes down exponentially in $n$. Here $mathrm{eff}^*(mathsf{V}, p)$ is a distributional version of the quantum efficiency or partition bound introduced by Laplante, Lerays and Roland (2014), which is a lower bound on the distributional quantum communication complexity of computing a single copy of $mathsf{V}$ with respect to $p$. As an application of our result, we show that it is possible to do device-independent quantum key distribution (DIQKD) without the assumption that devices do not leak any information after inputs are provided to them. We analyze the DIQKD protocol given by Jain, Miller and Shi (2017), and show that when the protocol is carried out with devices that are compatible with $n$ copies of the Magic Square game, it is possible to extract $Omega(n)$ bits of key from it, even in the presence of $O(n)$ bits of leakage. Our security proof is parallel, i.e., the honest parties can enter all their inputs into their devices at once, and works for a leakage model that is arbitrarily interactive, i.e., the devices of the honest parties Alice and Bob can exchange information with each other and with the eavesdropper Eve in any number of rounds, as long as the total number of bits or qubits communicated is bounded.
We propose a new point of view regarding the problem of time in quantum mechanics, based on the idea of replacing the usual time operator $mathbf{T}$ with a suitable real-valued function $T$ on the space of physical states. The proper characterization of the function $T$ relies on a particular relation with the dynamical evolution of the system rather than with the infinitesimal generator of the dynamics (Hamiltonian). We first consider the case of classical Hamiltonian mechanics, where observables are functions on phase space and the tools of differential geometry can be applied. The idea is then extended to the case of the unitary evolution of pure states of finite-level quantum systems by means of the geometric formulation of quantum mechanics. It is found that $T$ is a function on the space of pure states which is not associated to any self-adjoint operator. The link between $T$ and the dynamical evolution is interpreted as defining a simultaneity relation for the states of the system with respect to the dynamical evolution itself. It turns out that different dynamical evolutions lead to different notions of simultaneity, i.e., the notion of simultaneity is a dynamical notion.
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